Ophthalmic lenses (also: “OLs”) correct for optical errors of the eye, for example refractive errors such as myopia, hyperopia and presbyopia as well as other errors such as astigmatism. OLs can be positioned outside the human eye as, for example, spectacle lenses or contact lenses, which OLs can be monofocal (with one focal distance), multifocal (with multiple focal distances) or progressive (with a range of focal distances). Monofocal OLs provide sharp vision at a single focal distance and are most common, for example, as reading spectacles and most contact lenses. Multifocal OLs provide at least two foci, for example, bi-focal spectacles and multifocal contact lenses, or, alternatively, provide a continuous range of focal distances, as in, for example, progressive spectacles.
Intraocular lenses (also: “IOLs”) are OLs which are positioned inside the human eye and which can be monofocal, multifocal, progressive and accommodating (with variable focus). The inventions set forth in the present document can be applied to all OLs, for example spectacles, for example bi-focal spectacles; IOLs will be used henceforth to illustrate the principles and embodiments of said lenses.
IOLs are generally implanted by a surgeon after removal of the natural lens. Monofocal IOLs, diffractive and refractive multifocal IOLs providing multiple (generally two or three foci) are most common. Accommodating IOLs, providing a variable focus and driven by the accommodative process of the eye, are in development.
IOLs can be monofocal, providing monofocality, having a single focal distance. Monofocal IOLs in combination with the natural optical elements of the eye allow to project a single sharp image of a single object plane on the retina. IOLs can also be multifocal, providing multifocality, having multiple distinct focal distances. With multifocal IOLs a mix of multiple sharp images of multiple object planes on the retina can be obtained. Additionally, the focal distance of the lens can be fixed at manufacturing, as in fixed focus monofocal IOLs, or as in fixed multifocal IOLs. Alternatively, the focal distances of the IOLs can be adjustable as in adjustable focus monofocal OLs (of which the single focal distance is adjustable) and adjustable multifocal IOLs (of which at least one focal distance is adjustable). Accommodating IOLs have a fixed focal distance at the resting state (in an emmetrope eye) and a variable focal distance at the accommodative state. In accommodation the focal distance of the IOL depends on the degree of accommodation of the eye. In adjustable accommodating IOLs the fixed focal distance of the lens is adjustable which is important for obtaining emmetropia.
Diffractive IOLs combine a plurality of diffractive zones to provide multifocality. Diffractive multifocal IOLs have a large number of steep transition zones which zones cause significant image degradation due to scattering of light. In some cases diffractive IOLs may project ghost images, i. e. unwanted diffractive orders, on the retina leading to serious disorders in visual perception.
Refractive multifocal IOLs do not result in image degradation caused by scattering because of smooth surfaces and a limited number of transition zones. Such refractive multifocal IOLs include designs with a plurality of optical zones as in US2006192919 and WO2007037690, designs with radial-symmetry as in WO0108605 and WO9206400, designs with aspheric optical surfaces and with sloping optical surfaces along the azimuth as in WO0203126 and DE10241208 and designs including a smooth cubic phase mask as in US2003225455.
Chiral Optical Surfaces
A chiral optical surface is a surface of an optical element providing chiral phase modulation of light. For example, a chiral surface of a refractive optical element made of a material with a constant refractive index is a chiral optical surface. In mathematical terms, a three-dimensional surface is defined to be chiral if it is not invariant under parity transformation. This means that the mirror image of the surface can not be mapped onto the original by any rotations and translations, see FIG. 2. Definitions of chirality are given, for example, by M. Petitjean (J. Math. Phys. 43, 4147-4157, 2002) and Salomon et al. (J. Mater. Chem. 25, 295-308, 1999), both documents are included in this document by reference. The degree of chirality can be quantified in terms of topological charge or continuous chirality measure.
Chiral optical surfaces, in the context of this document, are characterized by certain steepness in radial and azimuthal directions. The steepness can progress either linearly or non-linearly according to any function which does not break the required chiral symmetry of the surface. Chiral optical surfaces can also include, but not necessarily so, at least one transition zone, for example, as shown in FIG. 3.
A chiral optical surface can be constructed from virtually any optical shape including parabolic, spherical, prismatic shapes etc. For example, consider a cubic surface defined byz=SU(x,y)≡U(axy2+bx3/3)  (1)in the coordinate system OXYZ, see FIG. 1, with the Z axis along the optical axis; U is the surface steepness measured, for example, in [mm−2]; a and b are the dimensionless constants, usually a=b when the X- and Y-axes have equal scales. According to U.S. Pat. No. 3,305,294, a pair of such cubic elements (a=b), mutually shifted along the X axis, can produce a variable-focus lens. Applying parity transformations (x,y)→(−x,y) or (x,y)→(x,−y), it can be easily found that this surface is not chiral. However, a combination of the two cubic surfaces can be made chiral, for example, a composite surface defined by
                    z        =                  {                                                                                                                S                                              U                                                  1                          ⁢                                                                                                                                                                      ⁡                                          (                                                                        x                          +                                                      x                            0                                                                          ,                        y                                            )                                                        ,                                      y                    ≥                    0                                                                                                                                                                  S                                              U                                                  2                          ⁢                                                                                                                                                                      ⁡                                          (                                                                        x                          -                                                      x                            0                                                                          ,                        y                                            )                                                        ,                                      y                    <                    0                                                                                                          (        2        )            where x0 is the constant of shift, U1 and U2 are the surface steepness parameters (generally U1≠U2), is a chiral surface. The surface defined by Eq. (2) is shown in FIG. 4.
By analogy with US2003225455, which describes one cubic surface on one optical element for an extended depth of field (EDF) intraocular lens, it can be noted that the chiral composite surface defined by Eq. (2) also provides continuous multifocality, or EDF. The multifocality ranges are determined by the parameters U1 and U2 in combination with x0 and can be chosen different providing two distinct multifocality zones (along the optical axis). Similarly, with an optical element comprising three cubic surfaces three distinct multifocality zones can be obtained etc.
The man skilled in the art can easily conclude that the combination (as in U.S. Pat. No. 3,305,294) of an optical surface defined by Eq. (2) with a cubic optical surface given by Eq. (1), as shown in FIG. 5, results in a lens with two distinct foci. These foci can be separated along the optical axis, Z-axis, and in the XY-plane (when linear tilt, i. e. wedge prism, is added to one of the optical surfaces).
Mathematical Framework
Assuming that a pair of identical optical elements is superimposed to form a variable-focus lens centred at the point O′ and, in cylindrical coordinates, each of the elements is specified by the functionz=S(r,θ),  (3)where z is the thickness of the element, r is the radius and θ is the polar angle and the point O′ characterized by the vector r0 is the centre of the optical surface with the area D, see FIG. 1. Let O be the centre of rotation, then if one element is rotated by +Δθ and another element is rotated by −Δθ the resulting thickness becomesΔz=S(r,θ+Δθ)−S(r,θ−Δθ).  (4)Taking the rotation-dependent thickness of a resulting variable-focus lens in the formΔz=ΔθA(r−r0)2,  (5)i. e. the optical power of the lens changes linearly with Δθ, A being the amplitude coefficient, applying Taylor expansions to Eqs. 4 and 5 it can be found
                                                                        Δ                ⁢                                                                  ⁢                ϑ                ⁢                                                                  ⁢                                                      A                    ⁡                                          (                                              r                        -                                                  r                          0                                                                    )                                                        2                                            =                            ⁢                                                S                  ⁡                                      (                                          r                      ,                                              ϑ                        +                        Δϑ                                                              )                                                  -                                  S                  ⁡                                      (                                          r                      ,                                              ϑ                        -                        Δϑ                                                              )                                                                                                                          ≅                            ⁢                                                2                  ⁢                                                                          ⁢                  Δϑ                  ⁢                                                            ∂                                              S                        ⁡                                                  (                                                      r                            ,                            ϑ                                                    )                                                                                                            ∂                      ϑ                                                                      +                                  2                  ⁢                                                                                    (                                                  Δ                          ⁢                                                                                                          ⁢                          ϑ                                                )                                            3                                                              3                      !                                                        ⁢                                                                                    ∂                        3                                            ⁢                                              S                        ⁡                                                  (                                                      r                            ,                            ϑ                                                    )                                                                                                            ∂                                              ϑ                        3                                                                                            +                …                                                                        (        6        )            Note that if |θ|<<1 Eq. (5) simplifies toΔz=ΔθA(r−r0)2≡ΔθA(r2+r02−2rr0[1−θ2/2]).  (7)The approximation of the unknown function S(r,θ) at |θ|<<1 can be determined from the differential equation
                                          ∂                          S              ⁡                              (                                  r                  ,                  ϑ                                )                                                          ∂            ϑ                          =                              A            2                    ⁢                                    (                                                r                  2                                +                                  r                  0                  2                                -                                  2                  ⁢                                                                          ⁢                                                            rr                      0                                        ⁡                                          [                                              1                        -                                                                              ϑ                            2                                                    /                          2                                                                    ]                                                                                  )                        .                                              (        8        )            The general solution of Eq. (8) takes the form
                                          S            ⁡                          (                              r                ,                ϑ                            )                                =                                                    A                2                            ⁢                                                (                                      r                    -                                          r                      0                                                        )                                2                            ⁢              ϑ                        +                                                            Arr                  0                                6                            ⁢                              ϑ                3                                      +            C                          ,                            (        9        )            where C is the integration constant. Using Eq. (9) the resulting thickness given by Eq. (4) becomes
                              Δ          ⁢                                          ⁢          z                =                              Δϑ            ⁢                                                  ⁢                          A              ⁡                              (                                                      r                    2                                    +                                      r                    0                    2                                    -                                      2                    ⁢                                                                                  ⁢                                                                  rr                        0                                            ⁡                                              [                                                  1                          -                                                                                    ϑ                              2                                                        /                            2                                                                          ]                                                                                            )                                              +                                                                                          (                    Δϑ                    )                                    3                                ⁢                A                            3                        ⁢                                          rr                0                            .                                                          (        10        )            The residual term (Δθ)3 Arr0/3 is a cone with a vertex at the origin O. The steepness of the cone changes cubically along with Δθ.
Consider now an extreme case when the centre of rotation O is located at infinity, or |r|,|r0|→∞. In this case the rotation is equivalent to a linear shift which is convenient to represent in Cartesian coordinatesx=r sin θ→rθy=(r−r0)cos θ→(r−r0)′  (11)Eq. (9) to take the form
                                                        z              =                            ⁢                                                S                  ⁡                                      (                                          r                      ,                      ϑ                                        )                                                  →                                                      S                    ~                                    ⁡                                      (                                          x                      ,                      y                                        )                                                                                                                          =                            ⁢                                                                    A                                          2                      ⁢                                                                                          ⁢                                              r                        0                                                                              ⁢                                      (                                                                                            x                          3                                                /                        3                                            +                                              xy                        2                                                              )                                                  +                                  C                  .                                                                                        (        12        )            which coincides with the main term of cubic surfaces described in U.S. Pat. No. 3,305,294 by L. Alvarez. A pair of Alvarez elements, being reciprocally displaced, produces a variable-focus parabolic lens with the optical power changing linearly with the lateral shift.
In another extreme, when the centre of rotation O coincides with the lens centre O′, or r0=0, Eq. (9) simplifies to
                                          S            ⁡                          (                              r                ,                ϑ                            )                                =                                                    A                2                            ⁢                              r                2                            ⁢              ϑ                        +            C                          ,                            (        13        )            and the resulting angle-depended thickness, as defined by Eq. (4), becomesΔz=ΔθAr2.  (14)
Equation (13) determines the thickness of a parabolic screw-type chiral optical element. In the simplest configuration when one surface of the optical element is flat, another surface is a parabolic screw-type chiral surface, or alternatively, a parabolic chiral optical surface, as illustrated in FIG. 3.
It should be noted that implementations of adjustable refractive power from rotation have been described in a prior art document “Adjustable refractive power from diffractive moiré elements,” by S. Bernet and M. Ritsch-Marte, Appl. Optics 47, 3722-3730 (2008), which document is included in the present document by reference. However, the authors limited the study to diffractive optical elements (DOEs) only. Their design resulted in a varifocal Fresnel lens with an additional sector lens of a different optical power. An optimized DOE design was suggested to avoid the additional sector lens.
As seen from Eq. (14), the optical power of a variable-focus lens centered at O′ is proportional to ΔθA and changes linearly with the angle of rotation Δθ. However, this expression is valid only for the angular sector Δθ≦θ<2π−Δθ. The sectors 0≦θ<Δθ and 2π−Δθ≦θ<2π result in an optical power proportional to ΔθA−πA. So, the variable-focus lens with two mutually rotated screw-type chiral optical elements produces two distinct foci, see FIGS. 7-9. Note also that the light intensities in the foci are proportional to 2(π−Δθ) and 2Δθ respectively.
For example, using formulas from W. J. Smith, Modern Optical Engineering, 3-rd. ed. (McGraw-Hill, New York, 2000), the optical power Φ, i. e. inverse focal length, of the lens combination comprising two mutually rotated screw-type chiral optical elements made of a material with the index of refraction n becomesΦ1=2(n−1)ΔθA,  (15)when Δθ≦θ<2π−Δθ andΦ2=2(n−1)(Δθ−π)A,  (16)when 0≦θ<Δθ and 2π−Δθ≦θ<2π.
It can be proven mathematically that a single chiral optical element with the thickness function according to Eq. (13) produces an effective multifocality, i. e. EDF. Making use of the general expression for the optical transfer function (OTF) of an incoherent optical system, i. e. an eye with an implanted chiral optical element, see J. W. Goodman, Introduction to Fourier Optics, (Roberts & Company, 2005), in the paraxial approximation it can be easily found thatH(ωr,ωα,φ)≅H(ωr,ωα+2φ/A,0),  (17)where H is the defocused OTF, φ is the defocus parameter (see J. W. Goodman for explanations), and ωr and ωα are the spatial frequency in polar coordinatesωx=ωy cos ωα,ωx=ωy sin ωα,  (18)where ωx and ωy are the corresponding spatial frequencies in the Cartesian coordinates. So, as seen from Eq. (17), defocusing does not lead to degradation of the resulting image (on the retina) but only rotation of the image. This rotation can be made very small by maximizing the steepness parameter A. FIG. 6 represents an ophthalmic lens comprising a single chiral optical element with continuous multifocality.